A high-performance analog Max-SAT solver and its application to Ramsey numbers
This work addresses combinatorial optimization problems like MaxSAT and Ramsey numbers, which are important in theoretical computer science and mathematics, but it is incremental as it builds on existing analog computing approaches.
The authors tackled the NP-hard MaxSAT problem by developing a continuous-time analog solver that uses the escape rate of dynamics to predict the global optimum, and applied it to find improved edge colorings for the Ramsey number R(5,5), achieving colorings with only two monochromatic 5-cliques on 43 vertices, the best result to date.
We introduce a continuous-time analog solver for MaxSAT, a quintessential class of NP-hard discrete optimization problems, where the task is to find a truth assignment for a set of Boolean variables satisfying the maximum number of given logical constraints. We show that the scaling of an invariant of the solver's dynamics, the escape rate, as function of the number of unsatisfied clauses can predict the global optimum value, often well before reaching the corresponding state. We demonstrate the performance of the solver on hard MaxSAT competition problems. We then consider the two-color Ramsey number $R(m,m)$ problem, translate it to SAT, and apply our algorithm to the still unknown $R(5,5)$. We find edge colorings without monochromatic 5-cliques for complete graphs up to 42 vertices, while on 43 vertices we find colorings with only two monochromatic 5-cliques, the best coloring found so far, supporting the conjecture that $R(5,5) = 43$.